Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5239,2,Mod(1,5239)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5239, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("5239.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5239 = 13^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5239.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(41.8336256189\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} - 17 x^{14} + 80 x^{13} + 98 x^{12} - 628 x^{11} - 158 x^{10} + 2458 x^{9} + \cdots - 147 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 403) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{16} - 4 x^{15} - 17 x^{14} + 80 x^{13} + 98 x^{12} - 628 x^{11} - 158 x^{10} + 2458 x^{9} + \cdots - 147 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 234985 \nu^{15} - 649739 \nu^{14} - 4442990 \nu^{13} + 11737815 \nu^{12} + 32793880 \nu^{11} + \cdots - 14255795 ) / 3154099 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 410788 \nu^{15} + 1274386 \nu^{14} + 8301681 \nu^{13} - 26138494 \nu^{12} - 65794620 \nu^{11} + \cdots - 99677416 ) / 3154099 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 526980 \nu^{15} + 1004839 \nu^{14} + 11285484 \nu^{13} - 18208549 \nu^{12} - 97120171 \nu^{11} + \cdots + 20992771 ) / 3154099 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 576101 \nu^{15} + 1321985 \nu^{14} + 12664367 \nu^{13} - 27267582 \nu^{12} - 111100193 \nu^{11} + \cdots - 92150633 ) / 3154099 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 784358 \nu^{15} - 1348140 \nu^{14} - 18214112 \nu^{13} + 27248288 \nu^{12} + 170196011 \nu^{11} + \cdots + 59706798 ) / 3154099 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 802740 \nu^{15} + 2267916 \nu^{14} + 16615333 \nu^{13} - 46241574 \nu^{12} - 136526765 \nu^{11} + \cdots - 178878045 ) / 3154099 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 806190 \nu^{15} + 2331414 \nu^{14} + 15709912 \nu^{13} - 44758234 \nu^{12} - 119944681 \nu^{11} + \cdots - 64556158 ) / 3154099 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 850612 \nu^{15} - 2605957 \nu^{14} - 15560111 \nu^{13} + 48167216 \nu^{12} + 108207470 \nu^{11} + \cdots + 34193170 ) / 3154099 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 878874 \nu^{15} - 2418510 \nu^{14} - 17786322 \nu^{13} + 47599181 \nu^{12} + 142637281 \nu^{11} + \cdots + 131774368 ) / 3154099 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1064679 \nu^{15} + 2903608 \nu^{14} + 21450107 \nu^{13} - 56574959 \nu^{12} - 170882518 \nu^{11} + \cdots - 96915048 ) / 3154099 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 1168224 \nu^{15} - 3355748 \nu^{14} - 22484283 \nu^{13} + 63618073 \nu^{12} + 168369826 \nu^{11} + \cdots + 71906220 ) / 3154099 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 1756897 \nu^{15} + 5326013 \nu^{14} + 33209620 \nu^{13} - 101451904 \nu^{12} - 242221543 \nu^{11} + \cdots - 153367298 ) / 3154099 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1970964 \nu^{15} - 5623664 \nu^{14} - 39099616 \nu^{13} + 109859647 \nu^{12} + 304896591 \nu^{11} + \cdots + 247630166 ) / 3154099 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} - \beta_{13} + \beta_{8} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + \beta_{14} + \beta_{12} + \beta_{10} + 8\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{15} + \beta_{14} - 7\beta_{13} + \beta_{9} + 8\beta_{8} + \beta_{6} + 2\beta_{2} + 28\beta _1 + 11 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{15} + 11 \beta_{14} + 8 \beta_{12} - \beta_{11} + 11 \beta_{10} + 2 \beta_{9} + \beta_{5} + \cdots + 78 \)
|
| \(\nu^{7}\) | \(=\) |
\( 67 \beta_{15} + 13 \beta_{14} - 42 \beta_{13} - \beta_{12} + 3 \beta_{10} + 17 \beta_{9} + 54 \beta_{8} + \cdots + 94 \)
|
| \(\nu^{8}\) | \(=\) |
\( 96 \beta_{15} + 94 \beta_{14} + 51 \beta_{12} - 15 \beta_{11} + 95 \beta_{10} + 34 \beta_{9} + \cdots + 476 \)
|
| \(\nu^{9}\) | \(=\) |
\( 476 \beta_{15} + 127 \beta_{14} - 241 \beta_{13} - 16 \beta_{12} - 5 \beta_{11} + 54 \beta_{10} + \cdots + 736 \)
|
| \(\nu^{10}\) | \(=\) |
\( 775 \beta_{15} + 736 \beta_{14} + 6 \beta_{13} + 301 \beta_{12} - 157 \beta_{11} + 754 \beta_{10} + \cdots + 3060 \)
|
| \(\nu^{11}\) | \(=\) |
\( 3345 \beta_{15} + 1112 \beta_{14} - 1352 \beta_{13} - 174 \beta_{12} - 97 \beta_{11} + 642 \beta_{10} + \cdots + 5537 \)
|
| \(\nu^{12}\) | \(=\) |
\( 6031 \beta_{15} + 5537 \beta_{14} + 131 \beta_{13} + 1697 \beta_{12} - 1410 \beta_{11} + 5750 \beta_{10} + \cdots + 20321 \)
|
| \(\nu^{13}\) | \(=\) |
\( 23516 \beta_{15} + 9190 \beta_{14} - 7419 \beta_{13} - 1610 \beta_{12} - 1213 \beta_{11} + 6380 \beta_{10} + \cdots + 40818 \)
|
| \(\nu^{14}\) | \(=\) |
\( 45979 \beta_{15} + 40805 \beta_{14} + 1821 \beta_{13} + 9164 \beta_{12} - 11647 \beta_{11} + \cdots + 137920 \)
|
| \(\nu^{15}\) | \(=\) |
\( 165995 \beta_{15} + 73275 \beta_{14} - 39425 \beta_{13} - 13668 \beta_{12} - 12464 \beta_{11} + \cdots + 297726 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.45635 | 2.80039 | 4.03363 | 3.48535 | −6.87871 | 3.54003 | −4.99531 | 4.84216 | −8.56123 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.21678 | −2.59178 | 2.91410 | 1.00571 | 5.74540 | −1.82096 | −2.02635 | 3.71732 | −2.22943 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.99449 | −1.58843 | 1.97798 | −2.66789 | 3.16810 | −1.73927 | 0.0439183 | −0.476895 | 5.32107 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.44015 | −2.79139 | 0.0740342 | 2.61757 | 4.02002 | 4.99660 | 2.77368 | 4.79185 | −3.76969 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.40506 | 1.52886 | −0.0258025 | −1.01618 | −2.14814 | 2.90958 | 2.84638 | −0.662589 | 1.42779 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.937316 | 1.68575 | −1.12144 | 0.290760 | −1.58008 | −3.85549 | 2.92577 | −0.158249 | −0.272534 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.327248 | −1.21221 | −1.89291 | −2.01769 | 0.396693 | −0.699230 | 1.27395 | −1.53055 | 0.660284 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.475674 | 0.345654 | −1.77373 | 0.161594 | 0.164419 | −4.07996 | −1.79507 | −2.88052 | 0.0768661 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.624841 | −2.78388 | −1.60957 | 0.619834 | −1.73948 | 2.35216 | −2.25541 | 4.74999 | 0.387298 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 1.01245 | −1.05136 | −0.974950 | 3.24438 | −1.06445 | −1.79572 | −3.01198 | −1.89463 | 3.28477 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 1.44935 | 0.0382575 | 0.100611 | −3.85977 | 0.0554484 | 1.66255 | −2.75288 | −2.99854 | −5.59415 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 1.48947 | 2.99747 | 0.218531 | −0.692562 | 4.46466 | 1.15260 | −2.65345 | 5.98485 | −1.03155 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.88981 | −0.327950 | 1.57140 | −1.75790 | −0.619765 | −0.652358 | −0.809976 | −2.89245 | −3.32211 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 2.44691 | 2.35527 | 3.98738 | 2.67528 | 5.76313 | 0.955539 | 4.86294 | 2.54728 | 6.54617 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.67507 | −1.67043 | 5.15599 | 2.86564 | −4.46851 | −5.21058 | 8.44248 | −0.209666 | 7.66579 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.71381 | 0.265781 | 5.36475 | −0.954130 | 0.721280 | 4.28452 | 9.13130 | −2.92936 | −2.58933 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(-1\) |
| \(31\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5239.2.a.l | 16 | |
| 13.b | even | 2 | 1 | 5239.2.a.k | 16 | ||
| 13.d | odd | 4 | 2 | 403.2.c.b | ✓ | 32 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.c.b | ✓ | 32 | 13.d | odd | 4 | 2 | |
| 5239.2.a.k | 16 | 13.b | even | 2 | 1 | ||
| 5239.2.a.l | 16 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\):
|
\( T_{2}^{16} - 4 T_{2}^{15} - 17 T_{2}^{14} + 80 T_{2}^{13} + 98 T_{2}^{12} - 628 T_{2}^{11} - 158 T_{2}^{10} + \cdots - 147 \)
|
|
\( T_{5}^{16} - 4 T_{5}^{15} - 31 T_{5}^{14} + 130 T_{5}^{13} + 339 T_{5}^{12} - 1490 T_{5}^{11} + \cdots - 163 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 4 T^{15} + \cdots - 147 \)
|
| $3$ |
\( T^{16} + 2 T^{15} + \cdots + 4 \)
|
| $5$ |
\( T^{16} - 4 T^{15} + \cdots - 163 \)
|
| $7$ |
\( T^{16} - 2 T^{15} + \cdots + 201937 \)
|
| $11$ |
\( T^{16} - 14 T^{15} + \cdots + 16 \)
|
| $13$ |
\( T^{16} \)
|
| $17$ |
\( T^{16} - 4 T^{15} + \cdots + 63559724 \)
|
| $19$ |
\( T^{16} - 22 T^{15} + \cdots - 86128439 \)
|
| $23$ |
\( T^{16} - 4 T^{15} + \cdots + 64364444 \)
|
| $29$ |
\( T^{16} + \cdots + 116300393492 \)
|
| $31$ |
\( (T + 1)^{16} \)
|
| $37$ |
\( T^{16} + \cdots - 5619651418284 \)
|
| $41$ |
\( T^{16} + \cdots + 641010277 \)
|
| $43$ |
\( T^{16} + \cdots - 2224783436 \)
|
| $47$ |
\( T^{16} + \cdots + 47289127696 \)
|
| $53$ |
\( T^{16} + \cdots - 15698233196 \)
|
| $59$ |
\( T^{16} + \cdots + 31361197805409 \)
|
| $61$ |
\( T^{16} + \cdots - 543516562532 \)
|
| $67$ |
\( T^{16} + \cdots - 749301251152 \)
|
| $71$ |
\( T^{16} + \cdots + 15553384173 \)
|
| $73$ |
\( T^{16} + \cdots - 7354897424 \)
|
| $79$ |
\( T^{16} + \cdots + 3031852028 \)
|
| $83$ |
\( T^{16} + \cdots - 2054567479152 \)
|
| $89$ |
\( T^{16} + \cdots + 918139278468 \)
|
| $97$ |
\( T^{16} + \cdots + 16757750670377 \)
|
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